3.131 \(\int \sqrt{1+x^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

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Rubi [A]  time = 0.0026484, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {195, 215} \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{1+x^2} \, dx &=\frac{1}{2} x \sqrt{1+x^2}+\frac{1}{2} \int \frac{1}{\sqrt{1+x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{1+x^2}+\frac{1}{2} \sinh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0042448, size = 18, normalized size = 0.86 \[ \frac{1}{2} \left (\sqrt{x^2+1} x+\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2] + ArcSinh[x])/2

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{{\it Arcsinh} \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)^(1/2),x)

[Out]

1/2*arcsinh(x)+1/2*x*(x^2+1)^(1/2)

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Maxima [A]  time = 1.39671, size = 20, normalized size = 0.95 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 1} x + \frac{1}{2} \, \operatorname{arsinh}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 + 1)*x + 1/2*arcsinh(x)

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Fricas [A]  time = 1.99301, size = 69, normalized size = 3.29 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 1} x - \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(x^2 + 1)*x - 1/2*log(-x + sqrt(x^2 + 1))

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Sympy [A]  time = 0.176005, size = 15, normalized size = 0.71 \begin{align*} \frac{x \sqrt{x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)**(1/2),x)

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

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Giac [A]  time = 1.05435, size = 34, normalized size = 1.62 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 1} x - \frac{1}{2} \, \log \left (-x + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^2 + 1)*x - 1/2*log(-x + sqrt(x^2 + 1))